A135110 -id:A135110 - cp's OEIS frontend

A037308 Numbers whose base-2 and base-10 expansions have the same digit sum.

0, 1, 20, 21, 122, 123, 202, 203, 222, 223, 230, 231, 302, 303, 410, 411, 502, 503, 1130, 1131, 1150, 1151, 1202, 1203, 1212, 1213, 1230, 1231, 1300, 1301, 1402, 1403, 1502, 1503, 1510, 1511, 2006, 2007, 2032, 2033, 2102, 2103, 2200, 2201, 3006, 3007, 3012

Offset: 1

Clark Kimberling

n is in the sequence iff n+(-1)^n is in the sequence. [Robert Israel, Mar 25 2013]

			122 is a member, since digital-sum_2(122) = 5 = digital-sum_10(122).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000120, A007953, A054899, A131451, A133620, A133900, A134599, A135110, A180018.

N:= 10000; # to get all elements up to N
select(x -> (convert(convert(x,base,10),`+`)-convert(convert(x,base,2),`+`)=0), [$0..N]); # Robert Israel, Mar 25 2013

Select[Range[0, 5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 10]] &] (* Jean-François Alcover, Mar 07 2016 *)

is(n)=hammingweight(n)==sumdigits(n); \\ Charles R Greathouse IV, Sep 25 2012

def ok(n): return sum(map(int, str(n))) == sum(map(int, bin(n)[2:]))
print(list(filter(ok, range(3013)))) # Michael S. Branicky, Jun 20 2021

[n for n in (0..10000) if sum(n.digits(base=2)) == sum(n.digits(base=10))] # Freddy Barrera, Oct 12 2018

From Reinhard Zumkeller, Aug 06 2010: (Start)
A007953(a(n)) = A000120(a(n));
A180018(a(n)) = 0. (End)

Edited by N. J. A. Sloane Nov 29 2008 at the suggestion of Zak Seidov

A135120 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 21, 222, 223, 1230, 1231, 1502, 2200, 2201, 3012, 3013, 10431, 12214, 12215, 12250, 12251, 14102, 15003, 15021, 16011, 20040, 20041, 22130, 23211, 23230, 23231, 24003, 30070, 30071, 30105, 30231, 30321, 31005, 31150, 31151, 31420

Offset: 1

Hieronymus Fischer, Dec 24 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3.

G. C. Greubel, Table of n, a(n) for n = 1..1100

Cf. A000040, A007953, A054899, A131451, A133620, A133900, A134599, A135110.

Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] ==  Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 26 2016 *)

is(n)=my(t=sumdigits(n)); t==hammingweight(n) && t==sumdigits(n,3) \\ Charles R Greathouse IV, Sep 26 2016

A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

Original entry on oeis.org

0, 1, 6, 7, 10, 11, 60, 61, 180, 181, 285, 300, 301, 575, 687, 754, 826, 827, 882, 883, 900, 901, 910, 911, 1254, 1305, 1311, 1326, 1327, 1335, 1377, 1383, 1386, 1387, 1395, 1431, 1506, 1507, 1532, 1626, 1627, 1650, 1651, 1890, 1891, 1955, 2013, 2036, 2040

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=6, since ds_2(6)=ds_3(6)=ds_5(6), where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0,3000],Length[Union[Total/@IntegerDigits[#,{2,3,5}]]]==1&] (* Harvey P. Dale, Sep 04 2014 *)

Added 0, Stanislav Sykora, May 06 2012

A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

Original entry on oeis.org

0, 1, 882, 883, 1386, 1387, 2502, 2503, 3453, 7555, 7652, 7665, 7931, 9751, 10101, 12250, 12251, 16893, 17010, 17011, 17515, 17550, 17551, 18285, 20301, 22050, 22051, 24406, 24407, 25053, 27503, 31654, 40930, 40931, 41951, 50878, 50879

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=882, since ds_2(882 )=ds_3(882 )=ds_5(882 )=ds_7(882 )=6, where ds_x=digital sum base x.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308, A212222.

Select[Range[0, 32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 7]] &] (* G. C. Greubel, Sep 27 2016 *)
Select[Range[0,51000],Length[Union[Total/@IntegerDigits[#,{2,3,5,7}]]] == 1&] (* Harvey P. Dale, Sep 18 2019 *)

Added 0, Stanislav Sykora, May 06 2012

A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

Original entry on oeis.org

1, 21, 261, 273, 17748, 17749, 20820, 20821, 65620, 65621, 70740, 70741, 83268, 83269, 86292, 86293, 1066068, 1066069, 1070420, 1135701, 1135893, 1135953, 5326161, 5330001, 5330241, 5330260, 5330261, 5506389, 5525829, 5526801, 5571909, 5574933, 5592321

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=21, since ds_2(21)=ds_3(21)=ds_10(21)=3, where ds_x=digital sum base x.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[5600000],Length[Union[Table[Total[IntegerDigits[#,n]],{n,2,4}]]]==1&] (* Harvey P. Dale, Aug 14 2013 *)

isok(n) = my(sd2=sumdigits(n, 2)); (sd2==sumdigits(n, 3)) && (sd2==sumdigits(n, 4)); \\ Michel Marcus, Aug 08 2018

a(31)-a(33) from Giovanni Resta, Aug 06 2018

A135123 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 6 all are equal.

Original entry on oeis.org

1, 12, 13, 114, 115, 366, 367, 477, 687, 864, 865, 876, 877, 1086, 1087, 1305, 1326, 1327, 1386, 1387, 1596, 1597, 1626, 1627, 1656, 1657, 1746, 1747, 1836, 1837, 1956, 1957, 2595, 2607, 2646, 2647, 3276, 3277, 3906, 3907, 3948, 3949, 4068, 4069, 5438

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=12, since ds_2(12)=ds_3(12)=ds_6(12), where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[5000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 26 2016 *)

A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.

Original entry on oeis.org

1, 64, 65, 4096, 4097, 4160, 4161, 262144, 262145, 262208, 262209, 266240, 266241, 266304, 266305, 16777216, 16777217, 16777280, 16777281, 16781312, 16781313, 16781376, 16781377, 17039360, 17039361, 17039424, 17039425, 17043456

Offset: 1

Hieronymus Fischer, Dec 31 2007, Dec 31 2008

Written as base 64 numbers the sequence is 1,10,11,100,101,110,111,1000,1001, ... (cf. A007088)

			a(7)=4161, since ds_2(4161 )=ds_4(4161 )=ds_8(4161 ), where ds_x=digital sum base x.

Michael De Vlieger, Table of n, a(n) for n = 1..8192
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Cf. A000695, A033045.

Select[Range[500000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 4]] == Total[IntegerDigits[#, 8]] &] (* G. C. Greubel, Sep 26 2016 *)
With[{k = 64}, Rest@ Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 28 2022 *)
Select[Range[171*10^5],Length[Union[Total/@IntegerDigits[#,{2,4,8}]]]==1&] (* Harvey P. Dale, May 14 2025 *)

a(n) = fromdigits(binary(n),64); \\ Kevin Ryde, Apr 02 2025

a(n) = (1/2)*Sum_{k=0..floor(log_2(n))} (1-(-1)^floor(n/2^k))*64^k.
G.f.: (1/(1-x))*Sum_{k>=0} 64^k*x^(2^k)/(1+x^(2^k)).
a(n) = A000695(A033045(n)) = A033045(A000695(n)). - Alan Michael Gómez Calderón, Mar 27 2025

Edited by N. J. A. Sloane, Jan 17 2009

A135125 Numbers such that the digital sum base 2 and the digital sum base 5 and the digital sum base 10 all are equal.

Original entry on oeis.org

1, 1300, 1301, 5010, 5011, 7102, 7103, 10050, 10051, 10235, 11135, 12250, 12251, 14015, 16102, 16103, 20060, 20061, 20206, 20207, 23230, 23231, 32012, 32013, 32302, 32303, 32410, 32411, 44000, 44001, 45010, 45011, 50012, 50013, 50300

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=1300, since ds_2(1300)=ds_5(1300)=ds_10(1300), where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..989

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[10000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 27 2016 *)

A135126 Numbers such that the digital sums in bases 3, 4, 5 and 6 all are equal.

Original entry on oeis.org

1, 2, 188, 668, 908, 1388, 1628, 2170, 2171, 2830, 2831, 3908, 4330, 4331, 6490, 6491, 8650, 8651, 10390, 10391, 10629, 12792, 12793, 12794, 17110, 17111, 17290, 17291, 25930, 25931, 36312, 36313, 36314, 37812, 37813, 37814, 41532, 41533, 41534

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(3)=188, since ds_3(188)=ds_4(188)=ds_5(188)=ds_6(188)=8, where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..1400

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[3000], Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 4]] ==  Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 6]] &] (* G. C. Greubel, Sep 27 2016 *)

A135128 Numbers such that the digital sums in bases 2, 3, 5 and 10 all are equal.

Original entry on oeis.org

1, 12250, 12251, 23230, 23231, 32410, 32411, 45010, 45011, 51130, 51131, 52030, 52031, 54010, 54011, 100053, 100090, 100091, 100305, 102250, 102251, 107002, 107003, 110134, 110170, 110171, 110350, 110351, 110460, 110461, 113050, 113051

Offset: 1

Hieronymus Fischer, Dec 31 2007

			a(2)=12250 since ds_2(12250 )=ds_3(12250 )=ds_5(12250 )=ds_10(12250 )=10, where ds_x=digital sum base x.

G. C. Greubel, Table of n, a(n) for n = 1..700

Cf. A007953, A054899, A131451, A133620, A133900, A134599, A135100, A135110, A135120, A037308.

Select[Range[32000], Total[IntegerDigits[#, 2]] == Total[IntegerDigits[#, 3]] == Total[IntegerDigits[#, 5]] == Total[IntegerDigits[#, 10]] &] (* G. C. Greubel, Sep 28 2016 *)
Select[Range[120000],Length[Union[Total/@IntegerDigits[#,{2,3,5,10}]]]==1&] (* Harvey P. Dale, Mar 30 2024 *)

cp's OEIS Frontend

A037308 Numbers whose base-2 and base-10 expansions have the same digit sum.

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A135120 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 10 all are equal.

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A135121 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 5 all are equal.

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A135127 Numbers such that the digital sums in bases 2, 3, 5 and 7 all are equal.

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A135122 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 4 all are equal.

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A135123 Numbers such that the digital sum base 2 and the digital sum base 3 and the digital sum base 6 all are equal.

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A135124 Numbers such that the digital sums in base 2, base 4 and base 8 are all equal.

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